American Institute of Mathematical Sciences

Advanced
2015, 2015(special): 1034-1040. doi: 10.3934/proc.2015.1034

Recurrence of multi-dimensional diffusion processes in Brownian environments

Hiroshi Takahashi 1, and  Yozo Tamura 2,

1. 

Colledge of Science and Technology, Nihon University, Narashino-dai, Funabashi 274-8501, Japan
2. 

Faculty of Science and Technology, Keio University, Hiyoshi, Yokohama 223-8522, Japan

Received  September 2014 Revised  February 2015 Published  November 2015

We consider limiting behavior of multi-dimensional diffusion processes in two types of Brownian environments. One is given values at different $d$ points of a one-dimensional Brownian motion, which is supposed to be a multi-parameter environment, and other is given by $d$ independent one-dimensional Brownian motions. We show recurrence of multi-dimensional diffusion processes in both Brownian environments above for any dimension and almost all environments. Their limiting behavior is quite different from that of ordinary multi-dimensional Brownian motion. We also consider cases of reflected Brownian environments.
Keywords: recurrence, transience., Diffusion process in Brownian environment.
Mathematics Subject Classification: Primary: 60K37, 60J60; Secondary: 60J6.
Citation: Hiroshi Takahashi, Yozo Tamura. Recurrence of multi-dimensional diffusion processes in Brownian environments. Conference Publications, 2015, 2015 (special) : 1034-1040. doi: 10.3934/proc.2015.1034
References:
[1]

T. Brox, A one-dimensional diffusion process in a Wiener medium., Ann. Probab., 14 (1986), 1206.   Google Scholar

[2]

M. Fukushima, S. Nakao and M. Takeda, On Dirichlet form with random date - recurrence and homogenization,, in, 1250 (1987), 87.   Google Scholar

[3]

K. Ichihara, Some global properties of symmetric diffusion processes., Publ. RIMS, 14 (1978), 441.   Google Scholar

[4]

K. Itô, On the ergodicity of a certain stationary process., Proc. Imp. Acad., 20 (1944), 54.   Google Scholar

[5]

K. Itô and H.P. McKean,Jr., "Diffusion Processes and Their Sample Paths,'', Springer-Verlag, (1965).   Google Scholar

[6]

D. Kim, Some limit theorems related to multi-dimensional diffusions in random environments,, J. Korean Math. Soc., 48 (2011), 147.   Google Scholar

[7]

G. Maruyama, The harmonic analysis of stationary stochastic processes,, Mem. Fac. Sci. Kyushu Univ., A4 (1949), 45.   Google Scholar

[8]

P. Mathieu, Zero white noise limit through Dirichlet forms, with application to diffusions in a random environment,, Probab. Theory Related Fields, 99 (1994), 549.   Google Scholar

[9]

Y. Sinai, The limit behavior of a one-dimensional random walk in a random environment,, Theory Probab. Appl., 27 (1982), 256.   Google Scholar

[10]

F. Solomon, Random walks in a random environment,, Ann. Probab., 3 (1975), 1.   Google Scholar

[11]

H. Takahashi, Recurrence and transience of multi-dimensional diffusion processes in reflected Brownian environments., Statist. Proba. Lett., 69 (2004), 171.   Google Scholar

[12]

H. Tanaka, Limit distributions for one-dimensional diffusion process in self-similar random environments,, in, 9 (1987), 189.   Google Scholar

[13]

H. Tanaka, Recurrence of a diffusion process in a multi-dimensional Brownian environment,, Proc. Japan Acad. Ser. A Math. Sci., 69 (1993), 377.   Google Scholar

show all references

References:
[1]

T. Brox, A one-dimensional diffusion process in a Wiener medium., Ann. Probab., 14 (1986), 1206.   Google Scholar

[2]

M. Fukushima, S. Nakao and M. Takeda, On Dirichlet form with random date - recurrence and homogenization,, in, 1250 (1987), 87.   Google Scholar

[3]

K. Ichihara, Some global properties of symmetric diffusion processes., Publ. RIMS, 14 (1978), 441.   Google Scholar

[4]

K. Itô, On the ergodicity of a certain stationary process., Proc. Imp. Acad., 20 (1944), 54.   Google Scholar

[5]

K. Itô and H.P. McKean,Jr., "Diffusion Processes and Their Sample Paths,'', Springer-Verlag, (1965).   Google Scholar

[6]

D. Kim, Some limit theorems related to multi-dimensional diffusions in random environments,, J. Korean Math. Soc., 48 (2011), 147.   Google Scholar

[7]

G. Maruyama, The harmonic analysis of stationary stochastic processes,, Mem. Fac. Sci. Kyushu Univ., A4 (1949), 45.   Google Scholar

[8]

P. Mathieu, Zero white noise limit through Dirichlet forms, with application to diffusions in a random environment,, Probab. Theory Related Fields, 99 (1994), 549.   Google Scholar

[9]

Y. Sinai, The limit behavior of a one-dimensional random walk in a random environment,, Theory Probab. Appl., 27 (1982), 256.   Google Scholar

[10]

F. Solomon, Random walks in a random environment,, Ann. Probab., 3 (1975), 1.   Google Scholar

[11]

H. Takahashi, Recurrence and transience of multi-dimensional diffusion processes in reflected Brownian environments., Statist. Proba. Lett., 69 (2004), 171.   Google Scholar

[12]

H. Tanaka, Limit distributions for one-dimensional diffusion process in self-similar random environments,, in, 9 (1987), 189.   Google Scholar

[13]

H. Tanaka, Recurrence of a diffusion process in a multi-dimensional Brownian environment,, Proc. Japan Acad. Ser. A Math. Sci., 69 (1993), 377.   Google Scholar

[1]

Yan Wang, Lei Wang, Yanxiang Zhao, Aimin Song, Yanping Ma. A stochastic model for microbial fermentation process under Gaussian white noise environment. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 381-392. doi: 10.3934/naco.2015.5.381
[2]

Dang H. Nguyen, George Yin. Recurrence for switching diffusion with past dependent switching and countable state space. Mathematical Control & Related Fields, 2018, 8 (3&4) : 879-897. doi: 10.3934/mcrf.2018039
[3]

Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control & Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401
[4]

Harish Garg. Novel correlation coefficients under the intuitionistic multiplicative environment and their applications to decision-making process. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1501-1519. doi: 10.3934/jimo.2018018
[5]

Shin-Ichiro Ei, Hiroshi Matsuzawa. The motion of a transition layer for a bistable reaction diffusion equation with heterogeneous environment. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 901-921. doi: 10.3934/dcds.2010.26.901
[6]

Henri Berestycki, Guillemette Chapuisat. Traveling fronts guided by the environment for reaction-diffusion equations. Networks & Heterogeneous Media, 2013, 8 (1) : 79-114. doi: 10.3934/nhm.2013.8.79
[7]

Genni Fragnelli, A. Idrissi, L. Maniar. The asymptotic behavior of a population equation with diffusion and delayed birth process. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 735-754. doi: 10.3934/dcdsb.2007.7.735
[8]

Arnaud Debussche, Sylvain De Moor, Julien Vovelle. Diffusion limit for the radiative transfer equation perturbed by a Wiener process. Kinetic & Related Models, 2015, 8 (3) : 467-492. doi: 10.3934/krm.2015.8.467
[9]

Harish Garg. Some robust improved geometric aggregation operators under interval-valued intuitionistic fuzzy environment for multi-criteria decision-making process. Journal of Industrial & Management Optimization, 2018, 14 (1) : 283-308. doi: 10.3934/jimo.2017047
[10]

Chengxia Lei, Jie Xiong, Xinhui Zhou. Qualitative analysis on an SIS epidemic reaction-diffusion model with mass action infection mechanism and spontaneous infection in a heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 81-98. doi: 10.3934/dcdsb.2019173
[11]

Danhua Jiang, Zhi-Cheng Wang, Liang Zhang. A reaction-diffusion-advection SIS epidemic model in a spatially-temporally heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4557-4578. doi: 10.3934/dcdsb.2018176
[12]

Isabelle Kuhwald, Ilya Pavlyukevich. Bistable behaviour of a jump-diffusion driven by a periodic stable-like additive process. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3175-3190. doi: 10.3934/dcdsb.2016092
[13]

Wuyuan Jiang. The maximum surplus before ruin in a jump-diffusion insurance risk process with dependence. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3037-3050. doi: 10.3934/dcdsb.2018298
[14]

Yejuan Wang, Peter E. Kloeden. The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4343-4370. doi: 10.3934/dcds.2014.34.4343
[15]

Donny Citra Lesmana, Song Wang. A numerical scheme for pricing American options with transaction costs under a jump diffusion process. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1793-1813. doi: 10.3934/jimo.2017019
[16]

Chihurn Kim, Dong Han Kim. On the law of logarithm of the recurrence time. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 581-587. doi: 10.3934/dcds.2004.10.581
[17]

Petr Kůrka, Vincent Penné, Sandro Vaienti. Dynamically defined recurrence dimension. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 137-146. doi: 10.3934/dcds.2002.8.137
[18]

Serge Troubetzkoy. Recurrence in generic staircases. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 1047-1053. doi: 10.3934/dcds.2012.32.1047
[19]

Michel Benaim, Morris W. Hirsch. Chain recurrence in surface flows. Discrete & Continuous Dynamical Systems - A, 1995, 1 (1) : 1-16. doi: 10.3934/dcds.1995.1.1
[20]

Philippe Marie, Jérôme Rousseau. Recurrence for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 1-16. doi: 10.3934/dcds.2011.30.1

 Impact Factor: 

Tools

Metrics

  • PDF downloads (79)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences